BPS算符与SO(N)超杨-米尔斯理论：plethysms与多米诺骨牌

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"N=4 $$ \mathcal{N} = 4 $$ SO(N)超Yang-Mills理论中的BPS算符：plethysms、多米诺骨牌和单词" 这篇学术文章深入探讨了在最大超对称的N=4 SO(N) Yang-Mills理论中的BPS算符构造。BPS（Bogomol'nyi-Prasad-Sommerfield）算符是一类特殊的量子场论算符，它们在特定条件下具有最小能量，是理论中的稳定态。在U(N)，SO(N)和Sp(N)规范群的背景下，这些算符可以通过置换和相关的代数结构来构建，这些结构允许构造出半BPS（拥有理论中一半对称性的状态）和四分之一BPS算符。作者详细研究了U(N)理论与SO(N)理论之间的定向投影映射，这涉及到Young图的基础。Young图是一种用于表示置换群的对称性的图形工具，在这里，它们被用来标记算符并确保它们在由自由场两点函数定义的内积下是正交的。具体来说，研究关注了U(N)理论中的Young图基础如何投影到SO(N)的半BPS算符上，发现这种映射与Littlewood-Richardson系数有简单联系。Littlewood-Richardson系数是用于计算两个Young图乘积得到的第三个图的系数，这里的特殊之处在于两个输入的图是相同的。文中还利用基于杨氏图的多米诺骨牌算法来处理这种多方面的改进，这种方法已被证明是可计算的。多米诺骨牌在这里作为可视化工具，帮助理解算符的组合结构。此外，对于SO(N)算符，特别是在半个和四分之一BPS扇区，通过考虑单词的组合来阐述生成函数的结构，这里的单词是指由基本字母组成的序列。在U(N)理论中，非周期性单词是Lyndon单词，而在SO(N)理论中，它们的类似物是通过最小周期条件定义的。文章进一步探讨了SO(N)重子扇区中Young图算符的正交基的归一化因子的计算，这是理解BPS算符相互作用和性质的关键部分。通过这些研究，学者们不仅深化了对N=4 SO(N)超Yang-Mills理论的理解，也提供了计算和分析此类理论中复杂算符的新方法。该研究发表于JHEP11(2018)035，由Springer出版，接收日期为2018年5月21日，接受日期为10月9日，最终发布日期为11月7日。作者Christopher Lewis-Brown和Sanjaye Ramgoolam分别来自英国伦敦玛丽女王大学的物理与天文学院和南非威特沃特斯兰德大学的物理学院及Mandelstam理论物理研究所。他们的电子邮件地址分别为c.h.lewis-brown@qmul.ac.uk和s.ramgoolam@qmul.ac.uk。

JHEP11(2018)035

3.3.2 Construction and counting

Since X and Y are anti-symmetric, X

⊗n

⊗ Y

⊗m

is anti-invariant under S

] × S

]

permutations, while the contractor C

(δ)

is invariant under S

n+m

]. Combining this anti-

invariance and invariance means that O

is invariant under

β 7→ (−1)

αβγ

−1

α ∈ S

n+m

] , γ ∈ S

] × S

] (3.9)

We have used γ

−1

rather than γ so that this forms an action of the direct product group

n+m

] ×(S

] × S

]), but the statement would have been entirely equivalent had

we used just γ instead. For simplicity, we will use γ for the remainder of this section. Using

this invariance and the fact that O

is linear in β, we have

(n+m)

(n + m)!

α∈S

n+m

]

n!2

γ∈S

]×S

]

sgn(γ)αβγ

= O

(3.10)

where

β =

2n+2m

(n + m)!n!m!

α∈S

n+m

]

γ∈S

]×S

]

(−1)

αβγ (3.11)

The elements which are invariant under (3.9) form a subspace of C(S

2n+2m

) that we call

n,m

. Since

β is invariant under this transformation, we see that without loss of generality,

mesonic operators are labelled by A

n,m

rather than the full algebra C(S

2n+2m

). Note that

we have put SO in the superscript rather than SO(N) because this subspace depends only

on the invariance (3.9) and is therefore independent of N.

It is well known that the group algebra C(S

) is isomorphic to the algebra of complex

functions on the group (with multiplication deﬁned by convolution). Explicitly, given a

function f : S

→ C, we can deﬁne a corresponding algebra element β =

σ∈S

f(σ)σ.

Conversely, given an algebra element β, the coeﬃcients in its linear expansion give us a

function f.

Under this isomorphism, A

n,m

maps to a subspace of the full space of functions. This

subspace consists of functions satisfying the equivalent (anti-)invariance to (3.9), given by

f(σ) = (−1)

f(ασγ)

We call this space F

n,m

To ﬁnd a basis of gauge-invariant operators we want to ﬁnd a basis for A

n,m

. Since

this is isomorphic to F

n,m

, we can equivalently ﬁnd a basis for the function space.

Given an arbitrary function f on the group, we can project it to F

n,m

by averaging

over the double coset, just as we did in (3.11). This mapping from an arbitrary function

to an (anti-)invariant one is surjective since if f is already (anti-)invariant, we have

f = f.

Thus we can produce a spanning set for F

n,m

by averaging a basis of ordinary functions.

An obvious choice for a basis would be the functions {f

: σ ∈ S

2n+2m

}, where f

evaluates to 1 on σ and 0 on any other permutation. These would lead to the basis of

multi-traces that we already considered at the end of section 3.3.1. Since we want an

– 14 –

JHEP11(2018)035

exactly orthogonal basis for the F

n,m

, we instead consider the matrix elements of the

irreducible representations

(σ) = hI, T |D

(σ)|J, T i (3.12)

where T ` 2n + 2m is a Young diagram, and {|T, Ii}, {|T, Ji} are two bases, not nec-

essarily the same, for the carrier space of T . Clearly I and J must take d

diﬀerent

values each, but these values may have more structure than just the numbers 1, 2, . . . , d

Averaging (3.12) gives

(σ) =

2n+2m

(n + m)!n!m!

α∈S

n+m

]

γ∈S

]×S

]

(−1)

(ασγ)

= D



[S]

n+m

σP

[A]

⊗[A]



= D



[S]

n+m



[A]

⊗ I



⊗ P

[A]



where we have used the projectors deﬁned in (2.8).

Now we see that a convenient choice for the bases {|T, Ii} and {|T, Ji} will be ones

that align nicely with the projectors P

[S]

n+m

and P

[A]

⊗[A]

respectively. This is most easily

done by aligning the bases with arbitrary representations of S

n+m

] and S

] ×S

By restricting the representation T of S

2n+2m

to S

n+m

], we get the decomposition

T =

[t]

T,[t]

[t]

Where [t] runs over the irreps of S

n+m

] and [t] occurs with multiplicity ν

T,[t]

. So the

new basis is



T, [t], µ

T,[t]

, I

[t]



Where 1 ≤ µ

T,[t]

≤ ν

T,[t]

counts which copy of [t] we are in and 1 ≤ I

[t]

≤dim[t] indexes the

basis vectors of [t].

For S

] × S

], we ﬁrst decompose the representation T of S

2n+2m

into a direct

sum of representations of S

× S

T =

R,S

R,S;T

R ⊗ S

where R runs over Young diagrams with 2n boxes, S runs over Young diagrams with 2m

boxes and the g

R,S;T

are the Littlewood-Richardson coeﬃcients. Then R and S can be

decomposed into S

] and S

] representations as we did above with T . Doing this,

we get the basis



T, R, S, λ, [r], µ

R,[r]

, I

[r]

, [s], µ

S,[s]

, I

[s]





R, λ, [r], µ

R,[r]

, I

[r]



⊗



S, λ, [s], µ

S,[s]

, I

[s]



where 1 ≤ λ ≤ g

R,S;T

indexes which copy of R ⊗ S we are in inside T . More formally,

λ indexes the basis vectors of the multiplicity space for R ⊗ S inside T . Choosing this

basis requires the use of an algebra deﬁned in section 3.5, so we delay reviewing this until

section 3.6.

– 15 –

JHEP11(2018)035

Therefore we can construct a spanning set for F

n,m

by considering the matrix

elements of



[S]

n+m

σP

[A]

⊗[A]



with respect to the S

n+m

] basis on the left and the S

] × S

] basis on the right.

Explicitly, they are

T,[t],µ

T,[t]

[t]

,R,S,λ,[r],µ

R,[r]

[r]

,[s],µ

S,[s]

[s]

(σ)



T, [t], µ

T,[t]

, I

[t]





[S]

n+m

σP

[A]

⊗[A]







R, λ, [r], µ

R,[r]

, I

[r]



⊗



S, λ, [s], µ

S,[s]

, I

[s]





(3.13)

It is a result from the representation theory of the wreath product [39, chapter VII.2]

that when an irrep R of S

is restricted to be a representation of S

], the completely

symmetric representation of S

] appears in the direct sum decomposition if and only if

R has an even number of boxes in each row, and then it appears with multiplicity 1. By

transposing the Young diagrams, the completely anti-symmetric representation of S

]

appears in the direct sum decomposition if and only if R has an even number of boxes in

each column, and then it appears with multiplicity 1. Therefore, using the adapted bases,

we can write the representatives of the projectors as



[S]

n+m



(

|T, [S]ihT, [S]| if T has rows of even length

0 otherwise



[A]

⊗[A]



R`2n

S`2m

R,S have columns

of even length

R,S;T

λ=1

|R, λ, [A]ihR, λ, [A]| ⊗ |S, λ, [A]ihS, λ, [A]|

Where we have dropped the µ

T,[S]

n+m

and I

[S]

n+m

indices because the multiplicity and

dimension of [S]

n+m

are 1 (similarly for [A]

and [A]

). Now we see that the vast majority

of the functions (3.13) vanish. They are only non-zero when

T ` 2n + 2m has even row lengths

[t] = [S]

n+m

R ` 2n and S ` 2m have even column lengths

[r] = [A]

[s] = [A]

As we already noticed, the µ multiplicities and I dimension indices have become trivial,

so we can drop those. Since [t], [r] and [s] are predetermined, we can also drop them.

Therefore the non-zero functions are indexed only by T, R, S, λ, with 1 ≤ λ ≤ g

R,S;T

T,R,S,λ

(σ) =

(2n + 2m)!

hT, [S]|D

(σ)



|R, λ, [A]i ⊗ |S, λ, [A]i



(3.14)

– 16 –

JHEP11(2018)035

where T, R, S satisfy the appropriate conditions. Note we have introduced a normalisation

factor for later convenience.

Since F

T,R,S,λ

are diﬀerent matrix elements of diﬀerent irreps of S

2n+2m

, they are

linearly independent in the full algebra of functions, and therefore they are linearly inde-

pendent in F

n,m

. Hence they form a basis.

There is an ambiguity in (3.14), stemming from the choice of basis vectors. We chose

|T, [S]i to be the invariant basis vector, but we could equally well have chosen |T, [S]i

−|T, [S]i (since representations of S

2n+2m

are real, we cannot have a complex phase) and

followed the exact same process. There is a similar ambiguity in the vectors |R, λ, [A]i and

|S, λ, [A]i. Since we are concerned with the function F , it is only the total sign of the matrix

element that needs to be determined. This can be done by selecting a permutation σ and

setting its matrix element to be positive (or negative). Note that this requires the matrix

element to be non-zero. F

T,R,S,λ

is then determined on the whole of S

2n+2m

. Unfortunately

ﬁnding a permutation with guaranteed non-zero matrix element for a given T, R, S, λ is not

easy, so for now we leave F

T,R,S,λ

deﬁned up to a sign.

From the basis (3.14) for F

n,m

, we get a basis for A

n,m

T,R,S,λ

σ∈S

2n+2m

T,R,S,λ

(σ)σ

(2n + 2m)!

σ∈S

2n+2m

hT, [S]|D

(σ)



|R, λ, [A]i ⊗ |S, λ, [A]i



σ (3.15)

There is a caveat before we proceed to a basis of gauge-invariant operators. Since N is ﬁnite,

an SO(N) vector index can only take a maximum of N distinct values. The construction

of Young diagram representations in V

⊗2n+2m

involves anti-symmetrising down each of the

columns. Thus if T has more than N rows (l(T ) > N), the associated operator will vanish.

This extra condition gives a subspace of A

n,m

(similarly for F

n,m

) relevant for constructing

operators at ﬁnite N, although the space remains unchanged if N ≥ n + m. Incorporating

the restriction l(T ) ≤ N in addition to those already in place on R, S and T , the basis of

operators is

T,R,S,λ

(2n + 2m)!

σ∈S

2n+2m

hT, [S]|D

(σ)



|R, λ, [A]i ⊗ |S, λ, [A]i



(δ)



⊗n

⊗m



(3.16)

In this section we have only proved that these operators span the space of (mesonic)

gauge-invariant observables. In section 8 we prove they are orthogonal under the two point

function, and therefore they are also linearly independent. Thus they do form a basis,

as claimed.

The operators (3.16) were presented in [26]. The normalisation there diﬀers from (3.16)

by a factor of

(2n+2m)!

(2n)!(2m)!

– 17 –

JHEP11(2018)035

From the labelling in (3.16), we know the number of linearly independent mesonic

operators for n X ﬁelds and m Y ﬁelds is

SO(N);δ

n,m

R`2n with even column lengths

S`2m with even column lengths

T `2n+2m with even row lengths

l(T )≤N

R,S;T

(3.17)

This counting of operators (and the baryonic counting (3.24)) can be obtained directly

from group integral formulae for the generating function of the quarter-BPS sector. This

calculation is given explicitly in [26].

We now check that (3.16) and (3.17) agree with the half-BPS results presented in [24].

To reduce the quarter-BPS objects to half-BPS ones, we set m = 0.

Since S ` 2m, it must now be the empty Young diagram. Therefore R and T must be

the same, and the Littlewood-Richardson coeﬃcient g

R,S;T

is just 1, so λ is ﬁxed as well.

Hence the half-BPS operators are labelled only by T , which must be a Young diagram with

even column and row lengths. Thus it must be constructed from 2×2 blocks, just as found

in [24]. Since T ` 2n, this can only occur when n is even.

Deﬁne t `

to be the Young diagram deﬁned by replacing each 2 ×2 block in T with

a single box. More formally, we have T = 2t ∪ 2t. Since t is unconstrained as a partition

, there are p





half-BPS operators when n is even, and none when n is odd.

Explicitly, by setting m = 0 in (3.16), the half-BPS operators are

(2n)!

σ∈S

hT, [S]|D

(σ)|T, [A]iC

(δ)



⊗n



(3.18)

It can be shown that the operators in [24] diﬀer from this by a factor of

hT, [S]|D

(ρ)|T, [A]i

where ρ = (1, 3)(5, 7) . . . (2n − 3, 2n − 1) is the same permutation we saw in (3.3).

This matrix element was calculated in [42], and is given by

hT, [S]|D

(ρ)|T, [A]i =

(2n)!

(3.19)

This is just a special case of the full result in [42], which gives this matrix element for any

permutation σ ∈ S

. We need to develop several ideas before we can present this, and it

is given in (4.24).

3.3.3 Resolving sign ambiguity

We noted earlier that (3.14) contained an ambiguity in the choice of basis vectors that

meant the functions F

T,R,S,λ

(and therefore O

T,R,S,λ

) were only deﬁned up to a minus sign.

The equality (3.19) allows us to resolve this ambiguity in the half-BPS sector. Since this

matrix element is always non-zero, for any T , we can choose it to be positive (and have

already done so implicitly in (3.19)).

– 18 –

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